Olympiad problems

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2015 Bangladesh Mathematical Olympiad, 2

[b][u]BdMO National Higher Secondary Problem 3[/u][/b] Let $N$ be the number if pairs of integers $(m,n)$ that satisfies the equation $m^2+n^2=m^3$ Is $N$ finite or infinite?If $N$ is finite,what is its value?

2002 Korea Junior Math Olympiad, 6

Tags: solutions , algebra

For given positive integer $a$, find every $(x_1, x_2, …, x_{2002})$ that satisfies the following: (1) $x_1 \geq x_2 \geq … \geq x_{2002} \geq 0$ (2) $0< x_1+x_2+…+x_{2003}<a+1$ (3) $ x^2_1+x^2_2+…+x^2_{2003}+9=a^2$

2020 Abels Math Contest (Norwegian MO) Final, 3

Tags: equation , algebra , solutions , real solutions

Show that the equation $x^2 \cdot (x - 1)^2 \cdot (x - 2)^2 \cdot ... \cdot (x - 1008)^2 \cdot (x- 1009)^2 = c$ has $2020$ real solutions, provided $0 < c <\frac{(1009 \cdot1007 \cdot ... \cdot 3\cdot 1)^4}{2^{2020}}$ .

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